The following is a model for HIV.
Infected cells = T
Concentration of viral particles = C

where is the rate constant for viral clearance.

1) Show that there are 2 possible types of critical points at the origin and one dividing case, and state the values of c which correspond to each case. In each case clearly state the stability of characteristics of the critical point at the origin.

2) If c = 5.5 what will the phase portrait of the system look like. (teacher said to look at the phase portrait non-uniformly, say take (-0.01, 0.01) * (-1,-1)).

3) Under treatment, the model changes to

What are the possible types of critical point at the origin in this model?

4) Introducing a new variable, which is the non-infected particles. The new system is

Prove that the eigenvalues of the linearized matrix are all negative in this case and find out what they are.
Why are the solutions to this no simply exponentials?

I know this is a massive question, so I am going to show my working in a separate post shortly.